Robust Control of Evolutionary Dynamics

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Thesis by

Vanessa D. Jonsson

In Partial Fulfillment of the Requirements

for the Degree of

Doctor of Philosophy

California Institute of Technology

Pasadena, California

2016

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c

2016

Vanessa D. Jonsson

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Acknowledgments

I am humbled by the support of my main advisor Richard Murray, who was the

principal reason for my success through the relatively daunting yet extraordinary

experience of my graduate research. He provided an incredible amount of intellectual

freedom and support to pursue my own ideas and research projects, despite their

tangential association to his own objectives. His intellectual enthusiasm, integrity

and ability to excel at the highest levels in multiple areas of engineering and science

has been the inspiration to push the limits of my own knowledge while having strong

personal principles as a researcher. I am indebted to my second advisor, David

Baltimore, for giving me the immense opportunity to work in his lab five years ago,

despite my lack of experience in experimental biology, and in particular with live

HIV. His belief in my capabilities, support and enthusiasm for my research project

catapulted my understanding of the power and limitations of engineering techniques

in the application to biomedical problems. This total immersion among experts in

virology and immunology was an incredibly unique and humbling experience. I would

like thank Pamela Bjorkman, whose invaluable advice and support was critical to the

success of several of my research projects. Her open-mindedness and willingness to

support my research ideas was a constant source of encouragement and validation. I

am indebted to her for providing such a welcoming environment. I also wish to thank

John Doyle for his constant support throughout my graduate studies and beyond and

for continually challenging and supporting me in the most difficult of times.

One of the professors that was most influential in the earlier years of my graduate

research was Rob Phillips, who opened up a new world of physical biology for me.

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dynamics of evolution. I am grateful for his commitment to excellent teaching. I am

also thankful to Christopher Snow and Justin Bois, who provided useful guidance in

the areas of computational chemistry and equilibrium thermodynamics. I am indebted

to my collaborators Anders Rantzer and Nikolai Matni, whose help was invaluable

in the development of the control theoretic algorithms in this thesis. I would like to

thank Anthony West who served as an informal advisor and whose knowledge of the

structural biology of HIV antibodies was central in my progress and understanding.

I am thankful for all the support of my former colleagues in both the Baltimore and

Bjorkman labs who provided immense technical support and mentoring. I appreciate

the help of Alex Sigal who initially took me under his wing and showed me how to

navigate HIV tissue culture. I am indebted to Collin Kieffer, who provided technical

guidance and help with advanced HIV protocols and to Michael Bethune for his help

and advice on experimental design and molecular biology protocols. I am thankful

for the helpful discussions with Ron Diskin, Louise Scharf, Jennifer Keeffe, and the

support from Geoffrey Lovely, Rachel Galimidi, Jocelyn Kim and Stella Ouyang. I

am grateful for my colleagues in the Murray group who have provided useful insight

and been supportive throughout the years. I thank Mark E. Davis at Caltech, for his

supportive mentoring and encouragement to avidly pursue my research ideas.

Finally, this journey would not have been possible without the endless support

of my wonderful family, my mother Ana Maria and father Russell, my stepmother

Patricia and my brother Russell. Above all, I want to thank my partner and best

friend Nikolai for his unwavering support – every moment of this journey was made

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Abstract

The application of principles from evolutionary biology has long been used to gain new

insights into the progression and clinical control of both infectious diseases and

neo-plasms. This iterative evolutionary process consists of expansion, diversification and

selection within an adaptive landscape — species are subject to random genetic or

epi-genetic alterations that result in variations; epi-genetic information is inherited through

asexual reproduction and strong selective pressures such as therapeutic intervention

can lead to the adaptation and expansion of resistant variants. These principles lie at

the center of modern evolutionary synthesis and constitute the primary reasons for

the development of resistance and therapeutic failure, but also provide a framework

that allows for more effective control.

A model system for studying the evolution of resistance and control of therapeutic

failure is the treatment of chronic HIV-1 infection by broadly neutralizing antibody

(bNAb) therapy. A relatively recent discovery is that a minority of HIV-infected

individuals can produce broadly neutralizing antibodies, that is, antibodies that

in-hibit infection by many strains of HIV. Passive transfer of human antibodies for the

prevention and treatment of HIV-1 infection is increasingly being considered as an

alternative to a conventional vaccine. However, recent evolution studies have

uncov-ered that antibody treatment can exert selective pressure on virus that results in the

rapid evolution of resistance. In certain cases, complete resistance to an antibody is

conferred with a single amino acid substitution on the viral envelope of HIV.

The challenges in uncovering resistance mechanisms and designing effective

com-bination strategies to control evolutionary processes and prevent therapeutic failure

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x

• Can we predict the evolution to resistance by characterizing genetic alterations

that contribute to modified phenotypic fitness?

• Given an evolutionary landscape and a set of candidate therapies, can we

com-putationally synthesize treatment strategies that control evolution to resistance?

To address the first question, we propose a mathematical framework to reason about

evolutionary dynamics of HIV from computationally derived Gibbs energy fitness

landscapes — expanding the theoretical concept of an evolutionary landscape

orig-inally conceived by Sewall Wright to a computable, quantifiable, multidimensional,

structurally defined fitness surface upon which to study complex HIV evolutionary

outcomes.

To design combination treatment strategies that control evolution to resistance,

we propose a methodology that solves for optimal combinations and concentrations

of candidate therapies, and allows for the ability to quantifiably explore tradeoffs in

treatment design, such as limiting the number of candidate therapies in the

combi-nation, dosage constraints and robustness to error. Our algorithm is based on the

application of recent results in optimal control to an HIV evolutionary dynamics

model and is constructed from experimentally derived antibody resistant phenotypes

and their single antibody pharmacodynamics. This method represents a first step

towards integrating principled engineering techniques with an experimentally based

mathematical model in the rational design of combination treatment strategies and

offers predictive understanding of the effects of combination therapies of evolutionary

dynamics and resistance of HIV. Preliminary in vitro studies suggest that the com-bination antibody therapies predicted by our algorithm can neutralize heterogeneous

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Chapter 1

Introduction

“The power of Selection, whether exercised by man or brought into play under nature through the struggle for existence and the consequent survival of the fittest, absolutely depends on the variability of organic beings. Without variability, nothing can be ef-fected; slight individual differences, however, suffice for the work, and are probably the chief or sole means in the production of new species." - Charles Darwin, 1868.

1.1

Background and Motivation

The modern synthesis. The core of current evolutionary theory was forged seventy

years after Charles Darwin’s On the Origin of Species [20], when statisticians and geneticists began laying the foundation for what is now called the ‘modern synthesis’.

This allowed the process of evolution to be described mathematically as the change

in frequencies of genetic traits in a population over time, uniting Darwin’s concept of

natural selection with a newly formed field of Mendelian genetics [30, 33, 103]. This

theoretical foundation and its corresponding quantitative methods provided support

to better understand the tenets of evolutionary theory — that variation arises through

random genetic mutation, is inherited by offspring and these together with natural

selection leads to adaptation and speciation.

Evolutionary biologists have since expanded upon the modern synthesis

frame-work, drawing concepts and methods from other fields. The discovery of DNA as

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2

information, became the driving force of evolutionary theory. It transformed the

no-tion that selecno-tion was associated with phenotypic trait variano-tion to being a funcno-tion

of variations driven by genetic mutations. Since then, genetic mapping for numerous

phenotypic adaptations across multiple organisms have been established — such as

the evolution of influenza virus resistance to antivirals [80], the adaptation of wing

shape in [74] and the genetic basis resistance of malaria in humans [34, 39].

Evolution and disease. More recently, the application of evolutionary concepts has

been extended to the study of both infectious and neoplastic disease, giving new

in-sights into their progression and clinical control [62, 71, 72]. The underlying processes

are equivalent to other evolutionary models — cells or virus are subject to random

genetic mutation that result in variations; genetic information is inherited through

asexual reproduction and strong selective pressures such as therapeutic intervention

can lead to the adaptation and expansion of resistant variants. Genetic instability

of neoplasms drives single base sequence changes, chromosomal rearrangements and

gene fusion, and these can confer a selective advantage when the resulting phenotype

exhibits both a proliferative advantage and a defect in DNA repair [55]. Non-genetic

adaptation and phenotypic plasticity can be induced via oncogene inhibition, enabling

the survival in cancer cells during initial therapy and thereby promoting residual

dis-ease. Recent studies in EGFR mutant lung adenocarcinoma reveal that NF-κB sig-naling is rapidly engaged upon initial EGFR inhibitor treatment to promote tumor

cell survival [7].

The 1987 FDA approval of azidothymidine (AZT), a nucleoside analog

reverse-transcriptase inhibitor (NRTI) for the treatment of chronic HIV infection, was one of

the first signs of therapeutic promise in the treatment of chronic human

immunod-eficiency virus type 1 (HIV-1) infection — this treatment significantly reduced viral

replication in patients and led to clinical improvements [29]. However, the ability

of HIV to rapidly evolve drug resistance was soon observed in patients treated with

AZT— the genetic basis for their resistance was explained with the existence of three

amino acid substitutions in the reverse transcription gene [50]. It was not until the

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a new class of protease inhibitors (PIs) [61], that when combined with two NRTIs,

introduced the concept of highly active antiretroviral therapy (HAART). The

treat-ment of chronic HIV infection with HAART is considered one of the great successes of

modern medicine in that it radically changed clinical outcomes successfully reducing

both patient viral loads to virtually “undetectable" levels, transforming HIV from a

fatal disease to one that is a manageable chronic illness [18].

The success of antiretroviral combination therapy in controlling evolution of HIV

has provided insight into how the evolutionary processes of other disease models could

be controlled for progression and management of therapeutic resistance [4, 94, 99].

With the recent introduction of targeted therapies for the treatment of certain cancers

[36], new questions surrounding the effectiveness of tailoring treatments to an

individ-ual patient’s tumor and its implications with respect to the emergence of driver

mu-tations and resistant phenotypes are being raised. Specifically, these small molecule

inhibitors and monoclonal antibodies exploit particular genetic addictions and

vulner-abilities of cancer cells, establishing an environment in which the occurrence of mildly

drug resistant cells can develop an evolutionary advantage over those for which the

therapy is targeted [23, 27, 41, 97]. Clonal expansion of these evolutionary

advan-tageous cells is exacerbated by the presence of considerable genetic intra-tumor

het-erogeneity already present in treatment-naive patients, contributing to resistance and

the need for principled approaches to the design of combination targeted treatment

strategies.

Toward a Principled Design of Treatment Strategies. These disease models

illustrate more generally the challenges in uncovering resistance mechanisms and

de-signing effective combination strategies to control evolutionary processes that lead to

resistance. Similar evolutionary processes are involved in the context of HIV-1 and

selection of resistant mutants with respect to broadly neutralizing antibody (bNAb)

therapy. In this thesis, we focus our attention to this particular application, but

the mathematical techniques that we propose are relevant to other infectious and

non-infectious diseases where growth, mutation and selection are central.

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exert selective pressure on virus resulting in the rapid appearance and evolution of

resistant mutant viral population [22, 47, 100]. In certain cases, complete resistance

to a bNAb is conferred with a single amino acid substitution on the viral envelope

(Env). To address HIV evolution in this context, combination bNAb therapy has

been proposed and shown to effectively control infection and suppress viral loads

below detection in murine models [22, 47, 100].

Advances in the identification and engineering of anti-HIV-1 antibodies have

pro-duced a large set of detailed molecular structures and neutralization data generated

against a broad panel of HIV-1 strains. Can we computationally predict evolution

to resistance by characterizing genetic alterations that contribute to modified

phe-notypic fitness? To address this question, we propose a computational model to

reason about evolutionary dynamics of HIV from computationally derived fitness

landscapes—linking the notion of genotype to phenotype in a quantifiable manner.

The second question we ask is, given a fitness landscape and a set of candidate

therapies, can we computationally synthesize treatment strategies and control

evo-lution? The rational design of combination antibody therapies for HIV treatment

involves the exploration of a large mutational space in the context of an ever growing

number of candidate antibodies — experimentally screening their combinations and

concentrations to effectively control evolution to resistance becomes increasingly

in-feasible. To address this, we require a scalable methodology that can take into account

increasing amounts of HIV/bNAb resistance data, bNAb pharmacodynamic models

and HIV mutational dynamics. To this end we propose a scalable and

computation-ally tractable algorithm that solves for optimal combinations and concentrations of

bNAbs to neutralize virus in light of viral evolution while simultaneously allowing

the designer to tailor treatment strategies in light of viral composition, maximum

achievable doses, number of bNAbs used and ability to support

pharmacodynam-ics/pharmacokinetic fluctuations, modeling and experimental error.

We briefly discuss prior computational work in addressing this problem, and follow

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1.2

Prior Work

Mathematical approaches. There have been numerous attempts to address

evo-lution of resistance with respect to therapeutic intervention from a mathematical

perspective within multiple disease contexts. Many computational results address

evolution to resistance by employing analytic methods and/or simulations on small

scale stochastic evolutionary dynamics models. The Michor lab [63] recently showed

the effects of different erlotinib dosing strategies in the presence of pharmacokinetic fluctuations on the evolution of resistance of non small cell lung cancer through

simu-lations of a stochastic evolutionary dynamics model. To address tumor heterogeneity

by rational combination therapy design, Zhao et al. [105], propose a static

multi-objective optimization formulation that is agnostic to evolutionary dynamics but that

models the effectiveness of independently acting, additive drugs on different initial

tumor populations. Proposed combination treatments were confirmed experimentally

for different tumor initial conditions in a murine lymphoma model [106].

The first models describing evolutionary processes in the context of HIV, were

inspired largely in part by Manfred Eigen’s original quasi species model [26] and have

since been proposed to study evolution, antigenic drift [70, 71]. Recent results by

Rosenbloom et al. [87], show that simulations of an evolutionary dynamics model

of HIV infection subject to changes in antiretroviral dynamics due to adherence are

consistent with clinical studies.

Control theoretic approaches. The challenge of designing treatment protocols

that prevent escape is one that has been addressed by control theoretic methods. For

cancer therapy, results in this spirit apply methods from optimal and receding horizon

control [3, 16], as well as gain scheduling [2], to synthesize treatment protocols that

are robust to parameter uncertainty, an inherent issue in all biological systems. In

the context of HIV and antiretroviral therapy, Hernandez-Vargas et al. [25] propose

a discrete time formulation that allows for the design of switching therapy strategies

to delay the emergence of highly resistant mutant viruses. There have been several

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tive control (MPC) to design optimal antiretroviral drug dosing strategies [56, 108].

Recent results in [84] and [8] consider a simplified bilinear model and the optimal

control problem is shown to be convex over a finite horizon for a predefined set of

initial states.

1.3

Thesis Contribution and Outline

In this thesis, we explore two questions in the evolutionary biology of disease: Can we

predict the evolution to resistance by characterizing genetic alterations that contribute

to modified phenotypic fitness? Given a fitness landscape and a set of candidate

ther-apies, can we computationally synthesize treatment strategies and control evolution?

We focus our application to the antibody treatment of chronic HIV infection, but the

mathematical techniques that we propose are relevant to other infectious and

non-infectious diseases. Many of the contributions of the following chapters are based on

a number of publications [42, 43, 44], indicated below.

Chapter 2: Evolutionary Dynamics on Computationally Derived Fitness

Landscapes. We propose a computational model to reason about evolutionary

dy-namics of HIV on computationally derived fitness landscapes. Our approach combines

well-utilized HIV dynamical systems models, incorporates infection and antibody

neu-tralization dynamics, a mutation process, and a method that uses energy minimization

calculations on structural information to quantify fitness differences between sensitive

and resistant strains. Specifically:

1. We propose and develop an extension of the least absolute shrinkage and

selec-tion operator (LASSO) to identify mutaselec-tional phenotypes and uncover potential

escape mutants from neutralizing anti HIV antibodies.

2. We develop a biophysical model based on Gibbs free energy of binding derived

from energy minimization calculations on structural information to quantify

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3. We develop an HIV evolutionary dynamics model to include infection and

neu-tralization reaction rates based on computed Gibbs energy fitness landscapes.

Chapter 3: Robust Control of Evolutionary Dynamics. Chapter 3 presents

three algorithms for the principled design of targeted combination drug treatment

strategies that explicitly account for the evolutionary dynamics of a generic disease

model, where the drugs under consideration are non-interacting and exhibit

inde-pendent additive effects. These algorithms allow the designer to quantifiably explore

tradeoffs between number of therapies used (controller sparsity), therapy

concentra-tions (magnitude of the gain) and ability to support pharmacokinetic fluctuaconcentra-tions

(robustness to perturbations). Our contribution specifically is itemized below.

1. Our first algorithm proposes a general iterative method that uses an H∞robust

control approach to design targeted combination therapy concentrations and is

effective in generating robustly stabilizing controllers.

2. Our second algorithm addresses large scale systems concerns lacking in the first

algorithm, presenting a scalable solution to the combination therapy problem

by reformulating it as a second order cone program (SOCP), with robustness

guarantees addressed by minimization of the induced L1 norm.

3. Our third algorithm solves the combination therapy problem subject to the

same design constraints (sparsity of the drug combination, maximum dosage and

robustness constraints) formulated as an SOCP while addressing the nonlinear

dynamics of individual drugs and of their combinations.

Chapter 4: Engineering Antibody Treatment Strategies to Control HIV.

We demonstrate our ability to control the evolution to resistance of HIV in the

pres-ence of antibody therapy, through the application of the combination therapy

algo-rithms developed in Chapter 3 as applied to experimental data derived from recent

published studies [14, 22, 47]. We also discuss a preliminary in vitro experimental methodology and results and show that the antibody treatment strategies synthesized

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8

EXPERIMENTAL+BIOLOGY+

Figure 1.1: Overview of the data, mathematical models and tools used in this thesis.

Section 3.5.3 controls infection despite the presence of a mixed initial population of

viruses, most of which are resistant to at least one antibody in the mix. Specifically:

1. We synthesize combination treatments and compare the respective H∞ and

the L1 combination therapy algorithms with respect to their performance and robustness to biologically relevant uncertainty models and unmodeled dynamics.

2. We develop a high throughputin vitro experimental system to identify the repli-cation and neutralization properties of HIV mutants and populate parameters

for our dynamical systems model, as well as test our predicted bNAb

combina-tion therapies.

3. We demonstrate successful in vitro validation of our computationally predicted bNAb combinations on heterogeneous viral populations comprised of resistant

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Chapter 2

Evolutionary Dynamics on

Computationally Derived Fitness

Landscapes

2.1

Introduction

In evolutionary biology, the concept of fitness landscape serves to associate

geno-type to some measure of fitness, or phenogeno-type. With the growing number of detailed

molecular structures and recent advances in modeling and computational approaches,

genotype-phenotype relationships are now being quantified and generated in an

au-tomated way. Thus, fitness landscapes are transitioning from a concept used for

visualization of fitness distributions, to computable, quantifiable, multidimensional

fitness surfaces upon which to study complex evolutionary outcomes.

Recent pre-clinical and clinical studies recently demonstrated that HIV can escape

from antibody mono therapy, and in some cases, combination therapy [14, 37, 47, 95].

In this chapter, we develop a computational framework that explains these

observa-tions and predicts the likelihood of certain resistant mutants in the presence of

anti-body mono therapy. Our method uses energy minimization calculations on structural

information and statistical inference on antibody neutralization data to quantify

fit-ness differences between sensitive and resistant strains and incorporates this data into

an HIV dynamical systems model of infection, mutation and antibody neutralization.

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HIV fitness landscapes have reasonable agreement with experimental findings. This

represents a first step in modeling and predicting HIV escape from antibody therapy

but has a broader application in evolutionary dynamics settings where a quantitative

relationship between genotype and evolutionary fitness can be established.

The chapter is organized as follows: in Section 2.2.1, we propose a statistical

in-ference method to uncover correlations between HIV-1 viral envelope (Env) sequence

data and antibody neutralization data. In Section 2.2.2, we compute HIV-1 fitness

landscapes using energy minimization techniques and argue that quantifying changes

to both infection and neutralization due to mutations are equally important aspects

in the study of HIV evolution to antibody resistance. In Sections 2.2.2 and 2.2.3, we

derive generalized Hill equations to express bound gp160/CD4 and gp160/antibody

as a function of the differences in Gibbs free energy of binding due to point mutations.

Section 2.2.4 connects preceding work by incorporating computed fitness landscapes

into a stochastic HIV evolutionary dynamical systems model of infection, mutation

and antibody neutralization and illustrates the applicability of these methods for the

prediction of resistance dynamics in light of antibody monotherapy for HIV.

2.2

Results

2.2.1

Statistical inference to uncover resistance phenotypes

We developed a statistical model and used this to uncover correlations between HIV-1

envelope (Env) sequence data and antibody neutralization data. In order to obtain a

model that could explain antibody neutralization data with a minimal set of residues

while taking into account irregularity in the experimental data set corresponding to

the limit of neutralization assays (Figure 2.1), we extended the well known least

absolute shrinkage and selection operator (Lasso) [98]. Our model, the saturated

Lasso (satlasso), returns a set of amino acid residues and their Env sequence

loca-tion that explains antibody neutralizaloca-tion data by minimizing error between model

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corresponding to saturated experimental data. A mathematical description of the

model can be found in Section 2.4.1. Model selection is performed with 5-fold cross

validation [79].

saturated( data( 8ANC195(neutraliza4on(

Virus(number(

A(

lo

g(

IC

50

)(

Figure 2.1: A. An example of the satlasso estimator as applied to 8ANC195 antibody neutralization data. Saturated data is due to the limit of the neutralization assay and is modeled in Equation (2.6). Red points correspond to experimental data, blue points correspond to the estimated model.

To assess the generalization of our satlasso model, we compare the first six largest

magnitude regressors to experimentally derived data sets and find that many Env

residues known to be critical to neutralization are identified (Figure 2.2). For the

gp120-gp41 bridge antibody 8ANC195, changes in glycosylation at 276, 234 and 230

sites are found to induce large changes in neutralization [91]. Specifically, the

intro-duction of a glycan at position 230 in a wild-type YU2 HIV strain, leads to a sixfold

increase in IC50, whereas removal of 234 and 276 sites leads to a more substantial

increase in IC50 [91, 101]. Our model identifies this epitope and captures the

corre-sponding changes in IC50 through the magnitude of each regressor 276++ (-2.79),

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V3

##

MP

ER#

gp

120-gp

41##

br

idg

e#

V2

#

CD4#

bs

#

Figure 2.2: The first six regressors with largest magnitude contributing to neutralization (blue) and resistance (red) for each of CD4 binding site, V2, V3, MPER and gp120-gp41 bridging antibodies in this study, solved with satlasso using 5-fold cross validation. Overlaid boxes correspond to amino acid changes that have been validated experimentally.

For all CD4 binding site antibodies in this study, we find that the existence of a Gly

at position 459 is most important for neutralization, confirming previous evolution

studies that mutations at position 459 confer resistance in 3BNC117 and

NIH-4546-G54W [47, 22]. Our model uncovers that an Arg at position 456 is also significant for

neutralization but less so than the site at position 459. Recently, Lynch et al. show

that a Trp mutation at position 456 has a modest effect on neutralization by many

CD4-bs antibodies with the exception of VRC-PG20 [57].

Functional studies on V3 loop binding antibodies 10-996 and 10-1074 reveal an

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critical site and shows that the existence of a His or Tyr at position 330 is necessary

for neutralization of 10-1074. There is some evidence that the H330Y substitution

has no effect on neutralization of either PGT121 and 10-1074 when this mutation was

tested on Simian-Human Immunodeficiency Virus, SHIVAD8 [89].

The highly conserved membrane proximal external region (MPER) of the gp41

transmembrane subunit on HIV-1 has recently been linked to epitopes of several

broadly neutralizing antibodies 2F5, 10E8 and 4E10 [38, 88, 65]. Alanine scanning,

structural and paratope analysis each indicate that 10E8 makes crucial contacts with

highly conserved residues W672, F673, W676 and K/R683 [38]. Although our model

does not recover these particular contact sites, it does identify a Thr substitution at

671 that is shown to raise IC80 values above 20 µg/ml in otherwise sensitive JR2 virus [38]. For 4E10, our model captures both one epitope site at 671 and a known

resistant substitution at 674S [9]. The 2F5 epitope is defined by a linear segment

of gp41 residues 662 - 668 with the key binding residues at N664, K665 and Y666

[6, 109]. Our model recovers the crucial binding site at position 665 and two sites on

the epitope, E662 and A667 that each confer resistance at the IC90 level with Ala

and Gly substitutions respectively [12, 109].

Our results demonstrate that satlasso recovers residues that are critical for

neu-tralization and those that contribute to resistance for a large class of anti-HIV-1

antibodies. Despite the fact that substitutions in certain positions can mediate

con-formational or other effects within Env that may be best represented by a nonlinear

model, some changes in neutralization can nonetheless be captured by this linear

regression model.

Recent studies in viral fitness costs associated with escape mutants from the class

of CD4-bs antibodies show that viral replicative fitness may be diminished with

cer-tain single mutations on the CD4 receptor binding site on gp120 [57], Chapter 4.

This suggests more broadly that evolution to resistance can be viewed as a function

of both viral replicative and antibody neutralization fitness. We elaborate this idea

in the following section and develop a method that quantifies both aspects of viral

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antibody monotherapy.

2.2.2

Gibbs Energy Landscapes Correlate With Known

Es-cape Mutations

We hypothesize that a virus’ capacity to infect and be neutralized by specific

neu-tralizing antibodies (NAbs) can be approximated by differences in Gibbs free energy

of binding associated to de novo point mutations on the envelope glycoprotein (Env) complexed with the CD4 receptor and antibody structure. To compute fitness

land-scapes relating to viral replication and antibody neutralization, we apply an empirical

force field, Fold X [93], to evaluate the effect of point mutations on the stability,

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decreased'ﬁtness'in'the'presence'of'an0body' A" C" B" in cre as ed '' an 0b od y'r es istan ce ' de cre as ed '' in fe c0 vity ' Increased'viral'ﬁtness'in'the'presence'of'an0body'

Figure 2.3: A. Gibbs binding energy differences computed between wild type and a subset of gp120 point mutations consisting of alanine substitutions on a subset of 3BCN117 antibody resistant mutations. Viral fitnessF_∆_gp_{/N Ab},is computed by Gibbs energy of binding differ-ences between neutralization and infection reactions and normalized with respect to largest value (Equation 2.3) (above). Gibbs energy differences for∆gp120/3BCN117 binding reac-tion (middle) and the ∆gp120/CD4 interaction (below). All mutations are numbered with respect to the HXBC2 reference genome. B. Viral fitness computed as inAfor CD4 binding site antibodies 3BNC117, 4546G54W, VRC01, VRC03, and VRCCH31. C.Fraction bound gp120/3BNC117 and CD4/gp120 as computed with Equations (2.5) using Gibbs energy landscapes for antibody and CD4 binding.

The resulting Gibbs free energy landscape provides an equilibrium

thermody-namic representation of the fitness of each point mutant with respect to the following

simplified binding reactions for infection and neutralization:

gp +cd4−→ gp·cd4, (2.1)

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where the HIV membrane glycoprotein gp, binds to the CD4 receptor, denoted cd4,

in reaction (2.1), and to antibody ` during neutralization in reaction (2.2).

We compute the difference in Gibbs free energy of binding between mutant and

wild type viral glycoprotein gp160,

∆∆G= ∆Gmut−∆Gwt,

for all point mutations on the solved portions of gp160 in both CD4 and antibody

complexes listed in Table 2.2, Section 2.5.

To quantify the effects of mutations on both infection and neutralization, we define

viral fitnessF∆gp/N Ab as a function of the cost of the mutation (∆gp) with respect to

antibody binding minus its cost with respect to CD4 binding

F∆gp/N Ab= ∆∆G(∆gp/NAb)−∆∆G(∆gp/CD4). (2.3)

To illustrate this measure of viral fitness, we compute binding energy landscapes

for CD4 binding site (CD4bs) antibody 3BNC117 using gp120/3BNC117 (PDB ID:

4JPV) and gp120/CD4 (PDB ID: 1G9N) structures and find reasonable agreement

between this measure and known resistant mutations in the presence of 3BNC117

[37, 57] (Figure 2.3, A). In particular, we note that the presence of either an Asp

at position 458 or an Ala at position 367 decreases binding to both 3BNC117 and

to CD4, simultaneously affecting the virus’ ability to infect and to be neutralized by

antibody. Binding energy differences between mutations and the CD4 receptor are

similar for both G367A and G458D (3.64 kcal/mol and 3.75 kcal/mol) whereas they

differ significantly for 3BNC117 binding (3.55 kcal/mol (G367A) and 11.15 kcal/mol

(G458D)), suggesting that both mutations could exhibit compromised viral replication

but that due to the greater difference in 3BNC117 binding, the Asp mutation at

location 458 is more likely to escape 3BNC117 neutralization. This is consistent with

our viral fitness calculations and model simulations (Figure 2.5, B) as well as previous

experimental validation showing that the Asp mutation at position 458 evolves in the

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To extend our analysis to other CD4bs antibodies, we compute Gibbs binding

en-ergy landscapes for 3BNC117, 4546G54W, VRC01, VRC03 and VRCCH31. Our viral

fitness calculations uncover that the most resistant substitutions across all CD4bs in

the study consist of an Asp substitution at either locations 458, 280 and 459 (Figure

2.3, B). Previous studies show that both the Asp substitutions at location 458 and

at location 280 evolve in 4546G54W and 3BNC117 monotherapy experiments [22, 37]

and abrogates neutralization by VRCCH31, and VRC01 [57]. The application of our

statistical inference model from Section 2.2.1 and previous experimental validation

[22, 37, 101] shows the importance of Gly at location 459 for effective neutralization

by all CD4bs antibodies.

More subtle differences in viral fitness differences can be uncovered by computing

Gibbs energy landscapes. We note that the Trp substitution at location 456 exhibits

a more modest increase in viral fitness (0.15-0.45) than the Asp substitution at 458

(0.45-1.0) across the CD4bs antibodies in the study, confirming recent experimental

studies that show that the Trp substitution at location 456 modestly decreases but

does not completely abrogate neutralization in all CD4bs antibodies [57].

2.2.3

Hill Functions Relate Gibbs Landscapes And Dynamical

Systems Parameters

Although analysis of detailed molecular structures may elicit molecular properties

by which resistance occurs, the pharmacodynamics of associated drug resistance are

less well understood. Therapeutic effect of resistant mutations is typically measured

as a change in IC50 relative to wild type. In the case of antiretroviral therapy for

HIV, Sampah et al. argue that inhibition can only be predicted if the shape of the

dose-response curve is known [90]. This relationship takes the form of a Hill function,

an equation extensively used in pharmacology to analyze nonlinear drug-receptor

relationships.

We derive generalized Hill equations and express the fraction of bound Env/CD4

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re-18 MOLECULAR)BINDING)MODEL) +" INFECTED) T)CELL) VIRUS) MUTATED) INFECTED) T)CELL) INFECTION) RATE) UNINFECTED) T)CELL) 0) 0) 0) NEUTRALIZED) VIRUS) +"

Y

" VIRUS) ANTIBODY) NEUTRALIZATION) RATE) 0) HIV)EVOLUTIONARY)DYNAMICS)MODEL) ANTIBODY) GIBBS)BINDING) ENERGIES) CD4) RECEPTOR) HIV) Correla*on"gibbs"and" inf"rate"" Correla*on"gibbs"and" neut"rate"" NEUTRALIZATION+DYNAMICS+ INFECTION+DYNAMICS+ A) B) C) MOLECULAR)BINDING)MODEL) +" INFECTED) T)CELL) VIRUS) MUTATED) INFECTED) T)CELL) INFECTION) RATE) UNINFECTED) T)CELL) 0) 0) 0) NEUTRALIZED) VIRUS) +"

Y

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Y

" VIRUS) ANTIBODY) NEUTRALIZATION) RATE) 0) HIV)EVOLUTIONARY)DYNAMICS)MODEL) ANTIBODY) GIBBS)BINDING) ENERGIES) CD4) RECEPTOR) HIV) Correla*on"gibbs"and" inf"rate"" Correla*on"gibbs"and" neut"rate"" NEUTRALIZATION+DYNAMICS+ INFECTION+DYNAMICS+ A) B) C)

A

C#

D#

B#

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actions, as a function of the differences in Gibbs free energy due to point mutations.

We begin by noting that the dissociation constant Kd is an equilibrium value that

can be expressed in terms of the free energy of binding, and define ∆Kmut

d to be the

ratio between mutant and wild type dissociation constants:

∆Kmut

d =

Kmut

d

Kwt

d

= ∆∆G/RT , (2.4)

whereR is the ideal gas constant and T is temperature. The resulting Hill equations model the fraction bound complexes involved in both infection and neutralization

reactions:

Rmut = [gp]

m

[gp]m₊₍_Kwt

inf∆Kinfmut)m

Nmut = [`]

n

[`]n₊₍_Kwt

neut∆Kmutneut)n ,

(2.5)

where the symbol [ ] indicates nanomolar concentration. ∆K_infmut and ∆K_neutmut are the difference between the dissociation constants for CD4 and antibody binding reactions

(2.1) and (2.2) between a mutationmutand the wild type viral glycoprotein as defined by Equation (2.4).

Remark 1. We assume that binding associated with the infection and neutralization process is noncooperative, and the corresponding Hill coefficientsmandnare approx-imately 1. Recent experimental studies characterizing antibody neutralization across

a diverse virus panel, suggests that for CD4 binding site antibodies n ≈ 0.9−1.37

[92], for V2 antibodies n ≈0.7 and for V3 binding antibodiesn ≈1.5. [64].

Figure 2.3 C depicts the Hill equations (2.5) associated with infection and

neu-tralization reactions for a subset of representative 3BNC117 resistant mutations and

their computed Gibbs binding energy landscapes. For highly resistant mutations most

likely to evolve in the presence of 3BNC117 like the Asp mutation at location 458, we

expect a small fraction of bound gp120/3BNC117 regardless of antibody concentration

and a moderate concentration of bound gp120/CD4, representing productive

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20

two orders of magnitude increase in 3BNC117 concentration to achieve an equivalent

fraction of gp120/antibody bound complexes and neutralization as wild type virus.

This suggests that at lower concentrations of antibody, viruses with similar fitness

profiles as A281T may not be all neutralized by antibody, creating an environment

in which this moderately resistant virus can become dominant or acquire additional

mutations and achieve greater resistance [22, 37, 47, 57].

2.2.4

Evolutionary Dynamics on Quantifiable HIV-1 Fitness

Landscapes

To reason about the dynamics of HIV evolution to resistance in the presence of

an-tibody therapy, we combined computed Gibbs energy landscapes with a stochastic

evolutionary dynamical system to model infection, mutation and antibody

neutral-ization (Figure 2.4, A, B). A mathematical description of the virus dynamics and

mutation model can be found in Section 2.4.1.

In order to validate of our model, we performed simulations of HIV infection and

antibody neutralization and compared these results to replication and neutralization

assays performed on gp120 mutations subcloned into a YU2-Env/NL4.3 infectious

backbone (Materials and Methods, Section 2.4.3). Both our simulations and

experi-ments were performed under the same conditions and were found to be in agreement,

(Figure 2.4, C, D), suggesting that our proposed mathematical model of HIV

infec-tion and neutralizainfec-tion based on Gibbs energy landscapes can be used to reasonably

predict both infectivity and neutralization rate changes due to point mutations on

Env. Simulations of our HIV infection model (Figure 2.4, C) confirm the well-known

exponential growth dynamics of HIV infection for mutations with small changes in

CD4 binding energies. For mutations with compromised CD4 binding such as the

Lys mutation on 279 and the Tyr mutation at 280, our simulations and experiments

show significantly decreased infection rates.

To illustrate how different neutralization profiles affect the evolution of

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of 4546G54W concentrations (Figure 2.4, D). We found that certain very resistant

mutations are less likely to be affected by high doses of 4546G54W, such as the Lys

mutation on 279 and the Tyr mutation at 280, whereas the degree of neutralization

of moderately resistant mutations such as the Ser substitution at 279 is dependent

on 4546G54W concentration (Figures 2.3, C and 2.4, D). High concentrations of

4546G54W are likely to shift the evolution of viral distributions towards highly

re-sistant mutations by creating an environment in which mutations with moderate

resistance are adequately neutralized, leaving the possibility of outgrowth of highly

resistant mutants despite low infectivity. We observe this phenomenon in our

evolu-tionary dynamics simulation in the presence of another CD4bs antibody, 3BNC117

(Figure 2.5,B). Specifically, the Lys substitution at location 280 is shown to outgrow

all other mutations with lower viral fitness, despite its lower infectivity (Figures 2.4,C

and 2.5,B).

To further explore how antibody concentration could influence the composition of

viral distributions at steady state, we ran fifty stochastic simulations of our HIV

evo-lutionary dynamics model for different concentrations of 3BNC117 (Figure 2.6,A). We

observe that the shape of the stationary distributions varies as a function of antibody

concentration with broader peaks occurring at lower antibody concentrations and

higher, narrower peaks at high antibody concentrations. We use the Gini coefficient

(Materials and Methods, ss:gini) to describe the shape of the stationary viral

distri-bution, and show that it increases as a function of different antibody concentration

for multiple CD4bs antibodies (Figure 2.6,B).

To understand which mutants might evolve in the presence of CD4bs antibodies,

we ran fifty evolutionary dynamics simulations using computed CD4 and antibody

binding energy landscapes for 1664 different mutants of YU2, using constant

concen-trations of either VRC01, VRC03, 3BNC117 or NIH4546. For a starting population

of monoclonal wild type virus, we show that the evolved mutations have close

agree-ment with previously studied escape mutations (Figure 2.5, C). Specifically, our HIV

evolutionary dynamics simulations in the presence of NIH-4546 reveal the evolution

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22

on antibodies NIH-4546 and NIH-4546G54W [47, 22]. Moreover, recent

experimen-tal results show that mutations N280D, N280K, A281T, G458D, G459D evolve from

3BNC117 monotherapy [37] - our simulations demonstrate the evolution of four out

of these five specific point mutations.

A B#

Figure 2.5: A. Fitness landscapes representing the difference in Gibbs binding energy be-tween wild type and mutations on the viral glycoprotein (∆gp120) and the CD4 receptor (PDB ID: 1G9N) (above) and the neutralizing antibody 3BNC117 (PDB ID: 4JPV) (below) for all amino acid substitutions on a subset of residues. Closed circles indicate binding site locations, stars indicate residue locations for which resistant mutations have been found.

B. One simulation of the HIV evolutionary dynamics model (Equation 2.7, Materials and Methods) on computed Gibbs energy landscapes shown in A and for a concentration of 5

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A" B"

C"

Figure 2.6: A. (Left) Fifty simulations of the HIV evolutionary dynamics model (Equation 2.7, Materials and Methods) on computed CD4 and antibody fitness landscapes for differ-ent NIH-4546 antibody concdiffer-entrations. Simulations are run with an initial condition of 105

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24

2.3

Discussion

Recent advances in the identification and engineering of anti-HIV-1 antibodies have

produced a large set of detailed molecular structures and neutralization data

gen-erated against a broad panel of HIV-1 strains. Recent computational analysis of

antibody neutralization data has been successful in categorizing antibodies with

re-spect to their neutralization activity [32], extracting the identities of Env residues that

are necessary for neutralization [101] and uncovering antibody epitopes [73]. Here,

we report a computational methodology that utilizes antibody neutralization data

and structural information to construct HIV fitness landscapes and reason about the

dynamics of resistance. It consists of the interpretation of neutralization data

us-ing statistical inference, the construction of fitness landscapes usus-ing computational

chemistry and the development of biophysical and mathematical model to capture

the dynamics of replication, mutation and selection.

Our statistical model is able to uncover critical Env residues involved in antibody

neutralization and is consistent with recent studies in antibody resistance. It does not

identify the entire structural epitope involved in the protein-protein contact. Rather,

satlasso identifies an epitope that is involved in the function of the protein-protein

interaction, in our case neutralization.

To address virus fitness in terms of its replication and neutralization capabilities,

we computed Gibbs binding energy landscapes on gp120/CD4 and gp120/NAb

struc-tures and found a good correspondence with our experimental studies. One of the

drawbacks of using force fields to compute fitness landscapes, is that they rely on

ex-perimental data and are therefore empirical. Furthermore, Xray structures are prone

to errors due in part to the non-physiological conditions under which the structure

is determined. Setting these concerns aside, tools like Fold X are geared specifically

toward screening the effect of single nucleotide polymorphisms (SNPs) on protein

sta-bility and are therefore well suited for our problem. For non-conservative mutations

(those not involving Ala or Gly scanning), a reorganization of the protein backbone is

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large structure reconfigurations such as a deletion or creation of a glycan structure, we

observe that Fold X is not always able to capture the corresponding effects on binding

energy. In the case where glycosylation changes have a large effect on neutralization

by antibody, satlasso is likely to recover these features, as illustrated by the accurate

uncovering of the glycan dependence of 8ANC195. Therefore, the combination of

data derived from satlasso and Gibbs energy landscapes offer complementary views

on viral fitness.

We present an HIV evolutionary dynamics model that is the first to our

knowl-edge to incorporate binding energy landscapes of replication and neutralization and

that accurately predicts the evolution of point mutations in the presence of anti-HIV

antibody monotherapy. Recent evolution studies uncovered the evolution of

dou-ble mutations that conferred resistance to several antibodies [22, 47, 37]. A clear

extension to our model is to incorporate double mutations in our Markov model of

mutational dynamics however a clear limitation is the accuracy of an empirical energy

minimization in the case of such double mutations.

2.4

Materials and Methods

2.4.1

Mathematical Models

satlasso. We define the saturated least absolute shrinkage and selection operator

(satlasso) and formulate it as a convex optimization problem. We consider that

the antibody neutralization data X = Xs+Xu is comprised of saturated data Xs,

corresponding to IC50s of very resistant viruses, and of unsaturated data Xu (Figure

2.1). Observe n predictor response pairs (xi, yi) where xi ∈ Rp and y ∈R. Forming

X ∈_Rn×p_, _X ₌_X

u+Xs with standardized columns, the saturated lasso, (satlasso)

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26

minimize β∈Rp

λ1

1

m||yu−Xuβ||2

| {z }

Error unsaturated data

+λ2

1

m

n

X

i=1

wi|βi|

| {z }

Sparsity

+λ3 max(ys−Xsβ,0)

| {z }

Error saturated data

, (2.6)

where wi = _βˆ1_i and βˆi = 1_ξ

Pξ

j=1|βˆ

j

i| where βˆ

j _{is the solution to the ordinary least}

squares problem for subset j.

Model selection was performed by 5 fold-cross validation. Our results exclude

mu-tations that are located in the signal peptide region, transmembrane and intracellular

regions corresponding to HXBC2-numbered residues (1-30), (685-706) and (706-end)

respectively. These locations on Env are not exposed to antibody binding and we

assume that are not subject to selective pressure by antibody.

Viral dynamics. To simulate how HIV might evolve resistance, we extended

the widely used HIV infection dynamics model [77] with a random mutation process

and included our Gibbs energy formulation to capture the effects of genomic variation

on the dynamics on infection and neutralization by antibody therapy. The stochastic

discrete time differential equation model of HIV evolution under antibody selection

is written as:

x[k+ 1] =λ+x[k]−(ηcPn_i Rix[k]−dxx[k])τ,

yi[k+ 1] =yi[k] + (ηcRix[k] + Υi[k]−dyyi[k])τ,

vi[k+ 1] =vi[k] + ((1−Ni)kvyi[k]−Nivi[k]−dvv[k])τ,

(2.7)

where x ∈ _R+ is the concentration of uninfected CD4+ T cells, yi ∈ R+ is the concentration of infected CD4+T cells that are actively producing mutant i,vi ∈R+ is the concentration of mutant virus i. Ri and Ni represent fraction bound ligand

associated with infection and neutralization of mutants, respectively as described in

Section 2.2.3. ηc = 104-105 is the number of CD4+ T cell receptors per uninfected

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Ri =

[gp]m

[gp]m_{+ (}_Kwt

inf∆Kinfi )m (2.8)

depends on the concentration of HIV glycoprotein for mutanti, and that is dependent on virus statevi. The glycoprotein concentration for mutants iat timekis calculated

by

[gp_i] =vi[k]

ηgp120

NA , (2.9)

where ηgp120 = 14 is the number of glycoprotein molecules per virion [17, 54] and NA

is the Avogadro constant. dx, dy, dv are degradation rates, respectively, of uninfected

CD4+ _{cells, infected CD4}+ _{cells, and virus.} _d

y and dv are assumed to be equivalent

for any mutant virus. λ is the T cell generation rate, and kv is the viral burst rate,

assumed to be equal for all mutants. Υi is the number of mutants igenerated by the

mutation process. Parameter values and units are listed in Table 2.4, Section 2.5.

Mutation process. The mutation process models the effects of error prone

reverse transcription allowing the genetic variability necessary for selection.

Experi-mental results indicate that single residue point mutations can cause resistance in the

presence of even the most potent broadly neutralizing antibodies [22, 47, 101], Chapter

4. Thus, we focus our analysis on point mutations on the viral envelope glycoprotein

but assume that mutations anywhere on the HIV genome are equally probable. We

do not consider mutations based on insertions, deletions or recombinations.

We allow for any number of single nucleotide changes to occur based on the reverse

transcription rate of mutation u = 3×10−5 _{mutations/base pair/replication cycle.} However, we track viruses that have at most one residue substitution from wild type

and assume that viruses with more than one residue substitution on contact sites are

considered unfit and disappear. If a second residue change occurs outside the contact

site, we assume this has the same fitness as the virus with one contact site residue

change. Based on these criteria, any particular virus in the system can be in one of

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28

or more mutations. Mutation dynamics are represented by a Markov process.

Let S = {wt, m1, ..., mn, nf, uf} be the state space associated with the Markov

process, corresponding to infected cell types where wt is the wild type infected cell,

m1, ..., mnare point mutant infected cells that produce virus with known fitness,nf is

an infected cell type that produces virus with unknown fitness, and uf is an infected cell type that produces virus considered unfit. As an approximation, we consider

infected cells that produce mutant virus where no fitness knowledge to be equivalent

in fitness to wild type, and those that produce virus with more than one residue

change are considered unfit. Let Xn be a random variable denoting the state of a

given cell at time n taking values in S, then its dynamics are given by the following state transition probabilities P ={pij}=P{Xn+1 =j|Xn=i}:

pwt→wt = (1−u)kl

pwt→mi =kcu(1−u)

kc−1₍₁−_u₎kh

pwt→nf =khu(1−u)kh−1(1−u)kc

pwt→uf = 1−pwt→wt−pwt→mi−pwt→nf

pmi→mi = 1− Pn

j6=ipmi→mj −pmi→uf

pmi→mj =

2

3kcpwt→mi

pmi→uf =pwt→nf,

(2.10)

for i ∈ {1, ..., n} and j 6=i, and where kl is the length of the entire HIV genome, kc

is the length of the genome for which fitness information exists, nc is the number of

possible point mutants with known fitness information,kh =kl−kc, anduis the rate

of mutation of HIV reverse transcriptase. All other state transition probabilities are

not considered. This Markov process at the single cell level induces a Markov process

at the population level.

2.4.2

Model Implementation and Simulations

Measuring the shape of viral distributions. The Gini coefficient measures

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M1# M2# …" MN#

MUT:#Known"Fitness"

UNFIT# WT#

MUT NF#

*"

**"pmi!mj

**" **"

**" *" *"

*"

pmi!mi

Figure 2.7: A finite state space Markov model representation of simplified mutation dy-namics. MUTF is an abstract state that encompasses all point mutations for which fitness information is known, MUTNF is an abstract state that defines all mutations for which fitness information is not known. The WT state is a wild type virus state, UNFIT final state is a state that has no infectivity and is considered unfit with respect to any selective pressure.

in a population. We use this measure as an approximation to the shape of the virus

distribution at steady state. The Gini coefficient G is calculated as

G=

Pn i=1

Pn

j=1|yi−yj|

2n2_µ , (2.11)

wheren is the number of all possible point mutations,yi is the percent population

of a point mutation i that occurs at steady state for all simulations and µ is their mean.

2.4.3

Experimental Methods

Mutagenesis, Virus Production and Cells. Site directed mutagenesis and

as-sembly PCR were used to generate YU2-NL43 Env mutants. YU2-NL43 was modified

using unique restriction sites EcoRI and XhoI. Inserts were generated by PCR using

primers EcoRI-CF (5’-GCCAGCCAGAATTCTGCAACAACTGCTGTTTATCCAT

TTCAG-3’) and

XhoI-CR-(5’-GCGTCGACCTCGAGATACTGCTCCCACCCCATC-3’) and individual sense and antisense mutagenesis primers corresponding to

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30

a 16 hour incubation at 30◦ C. Stocks were prepared using a DNA Midi kit (Zymo

Re-search). All gene constructs were verified by complete sequencing of gp160. Cell-free

virus was produced by transfection of HEK293T cells with YU2-NL43 virus coding

plasmid using BioT (Bioland Scientific). Viral supernatant was harvested at 48 h post

transfection, filtered through a 0.5µm filter and aliquots were stored at -80◦ C. Stock concentrations were quantified by p24 enzyme-linked immunosorbent assay (ELISA)

(Cell Biolabs). The YU2 Env/NL43 plasmid was obtained from the Nussenzweig

lab, Rockefeller University. The green fluorescent protein (GFP) reporter T-cell line

GXR-CEM is previously described in [11] was obtained through AIDS Research and

Reference Reagent Program, National Institute of Allergy and Infectious Diseases,

National Institutes of Health.

Protein Expression and Purification. Antibodies were transiently expressed

in HEK293T/17 cells or suspension HEK 293-6E cells (National Research

Coun-cil Biotechnology Research Institute, Montréal, QC, Canada) using 25-kDa linear

polyethylenimine (Polysciences) for transfection. Supernatants were passed over

Mab-Select SuRe protein A resin (GE Healthcare) or Protein G Sepharose 4 Fast Flow (GE

Healthcare) and eluted by using pH 3.0 citrate or glycine buffer, and then

immedi-ately neutralized. Antibodies were further purified by size exclusion chromatography

using a Superdex 200 or 75 10/300 GL column.

In Vitro Replication and Neutralization Assays. To initiate infection for

both replication and neutralization assays, GXR-CEM cells at 4×105 _{cells/ml were} infected with 200ngYU2-NL43 (HIV) virus stock in the absence of antibody and incu-bated for three days at 37◦ C. Two days after infection, uninfected GXR-CEM cells at

2×105 were pre-treated with 0, 0.02,0.08, 0.4,1.6, 6.4, 20, 80µg/ml of NIH4546G54W antibody. Three days after initial infection with cell free virus, the infected

GXR-CEM were washed and added to a final concentration of 1 %GFP-expressing donor

cells to uninfected pre-treated GXR-CEM cells. For the neutralization assay, a

con-stant concentration of antibody was maintained for each sample for three days

follow-ing secondary infection. Infection was determined by measurfollow-ing GFP reporter gene

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sec-ondary infection. Neutralization was determined by measuring the reduction in GFP

reporter gene expression in the presence of antibodies NIH4546G54W for three days.

Flow cytometry data was collected with a MACSQuant flow cytometer and

trans-formed to a python format using the FlowCytometryTools python package (Gore lab,

MIT). An automated gating and analysis software module written in python was

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32

2.5

Appendix

Antibody CVE λ1 λ2 λ3

3BNC117 0.387 5.05 3.25 1.7

3BNC55 0.3 5.05 4.6 0.35

NIH45-46G54W 0.612 1.45 1.45 7.1

45-46m2 0.46 1 4.15 4.85

45-46m7 0.425 2.35 6.4 1.25

VRC01 0.227 2.8 4.15 3.05

VRC03 0.316 30 60 10

VRC02 0.322 2.35 3.7 3.95

VRC-PG04 0.382 3.0 6.0 1.05

VRC-PG04b 0.382 50 40 10

b12 0.127 3.7 5.05 1.25

12A12 0.321 20 80 0

VRC-CH31 0.382 30 50 20

PG16 0.485 2.8 5.95 1.25

10-1074 0.373 1.9 3.25 4.85

10-996 0.486 4 5 1

PGT121 0.478 3.25 3.25 3.5

8ANC195 0.274 3.25 4.6 2.15

2F5 0.132 2.8 5.95 1.25

10E8 0.33 5.05 2.8 2.15

4E10 0.159 1.925 3.35 4.725

Table 2.1: 5-fold cross validation error (CVE), optimalλ1, λ2 andλ3 for satlasso models on

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Antibody Bound Virus PDB accession

NIH-4546G54W 93TH057 4JKP

NIH-4546 93TH057 3U7Y

VRC-01 93TH057 3NGB

VRC-CH31 93TH057 4LSP

3BNC117 93TH057 4JPV

VRC-PG04 93TH057 3SE9

VRC-03 93TH057 3SE8

PG16 YU2 4DQ0

PGT128 HXB2 3TYG

Table 2.2: Antibody and virus molecular structures and their Protein Data Bank (PDB) accession numbers.

Ligand Envelope molecule Kd Units Ref

CD4 Core gp120 220 nM [67]

CD4 full length gp120 22 nM [67]

12A12 YU2-gp140 0.3 nM [92]

3BNC60 YU2-gp140 6.81 nM [92]

3BNC117 YU2-gp140 6.54 nM [92]

3BNC55 YU2-gp140 57.8 nM [92]

8ANC195 YU2-gp140 34.3 nM [92]

45-46 YU2-gp140 6.75 nM [92]

VRC01 YU2-gp140 0.4 mM [92]

VRC-CH31 YU2 gp140 37.8 nM [107]

VRC-PG04 KER2018 core 11.9 nM [45]

VRC-03 YU2 gp140 16.1 nM [104]

PG16 V1-V2 complex glycan 1.26 mM [75]

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34

Parameter Value Units

λ 105 _cells_·_day−1

dx 0.01 day−1

dy 0.01 day−1

kv 100 day−1

ηCD4 105 CD4 molecules per T cell

ηgp120 14 gp120 molecules per virion

τ day

T 313 K

R 1.987 kcal· mol−1_·_K−1

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Chapter 3

Robust Control of Evolutionary

Dynamics

3.1

Introduction

A challenge inherent to the treatment of certain infectious and non-infectious diseases,

such as HIV or cancer, is the risk that the pathogen or tumor will evolve away

and become resistant to treatment methods that comprise the standard of care [35,

46, 48, 76]. Especially vulnerable to this phenomenon are treatment methods that

involve exposing the disease population (such as viruses or cancer cells) to therapies

targeting specific molecules involved in disease progression for an extended period

of time. While these targeted therapies have the benefit of allowing physicians to

tailor treatments to a patient’s tumor cell population, they nonetheless establish an

environment in which the occurrence of mildly drug resistant pathogens or tumor cells

can develop an evolutionary advantage over those for which the therapy is targeted

[23, 27, 41, 97], leading to so called “treatment-escape”.

The challenge of designing treatment protocols that prevent escape is one that

has been addressed by control theoretic methods. For cancer therapy, results in this

spirit apply methods from optimal and receding horizon control [3, 16], as well as

gain scheduling [2], to synthesize treatment protocols that are robust to parameter

uncertainty, an inherent issue in all biological systems. Zhao et al. [105] present

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36

problem for different initial tumor populations, when the drugs under consideration

have additive, linear effects on cell viability. Proposed combination treatments were

confirmed experimentally for different tumor initial conditions in a murine lymphoma

model [106]. In the context of HIV and antiretroviral therapy, the authors in [25]

propose a discrete time formulation that allows for the design of switching therapy

strategies to delay the emergence of highly resistant mutant viruses. Recent results

in [84] and [8], consider a simplified bilinear model and the optimal control problem

is shown to be convex over a finite horizon for a predefined set of initial states.

This chapter presents three algorithms for the principled design of targeted

combi-nation drug treatment strategies that explicitly account for the evolutionary dynamics

of a generic disease model, where the drugs under consideration are non-interacting

and exhibit independent additive effects. Our first algorithm, introduced in [42],

pro-poses a general iterative method that uses an H∞ robust control approach to design

targeted combination therapy concentrations and is effective in generating robustly

stabilizing controllers. Our second algorithm addresses the lack of scalability

symp-tomatic of semidefinite programming (SDP) formulations and proposes a scalable

solution to the combination therapy problem by reformulating it as a second order

cone program (SOCP), with robustness guarantees addressed by minimization of the

induced L1 norm. We also require that the synthesized controller be not only ro-bust to unmodeled dynamics, but also exhibit sparse structure and small feedback

gains. This is motivated by the fact that the number of therapies commonly used

in combination to treat a disease is often small while the number of potential usable

therapies is often very large [1]. Targeted therapies such as small molecule drugs or

antibodies exhibit a maximum effective concentration beyond which side effects are

likely to worsen and no additional drug benefits are seen. Our

Robust Control of Evolutionary Dynamics

Vanessa D. Jonsson

Contents

Abstract

Chapter 1

Prior Work

Thesis Contribution and Outline

EXPERIMENTAL+BIOLOGY+

Chapter 2

Results

Gibbs Energy Landscapes Correlate With Known

Hill Functions Relate Gibbs Landscapes And Dynamical

Evolutionary Dynamics on Quantifiable HIV-1 Fitness

Discussion

Materials and Methods

Model Implementation and Simulations

Experimental Methods

Appendix

NIH45-46G54W 0.612 1.45 1.45 7.1

VRC-PG04 93TH057 3SE9

Chapter 3